Calculus Examples

$\log (\log (100000^{200 x}))$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \log (x)$ and $g (x) = \log (100000^{200 x})$ .

Tap for more steps...

Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\log (100000^{200 x})$ .

$\frac{d}{d u_{1}} [\log (u_{1})] \frac{d}{d x} [\log (100000^{200 x})]$

Step 1.2

The derivative of $\log (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1} \ln (10)}$ .

$\frac{1}{u_{1} \ln (10)} \frac{d}{d x} [\log (100000^{200 x})]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\log (100000^{200 x})$ .

$\frac{1}{\log (100000^{200 x}) \ln (10)} \frac{d}{d x} [\log (100000^{200 x})]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \log (x)$ and $g (x) = 100000^{200 x}$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u_{2}$ as $100000^{200 x}$ .

$\frac{1}{\log (100000^{200 x}) \ln (10)} (\frac{d}{d u_{2}} [\log (u_{2})] \frac{d}{d x} [100000^{200 x}])$

Step 2.2

The derivative of $\log (u_{2})$ with respect to $u_{2}$ is $\frac{1}{u_{2} \ln (10)}$ .

$\frac{1}{\log (100000^{200 x}) \ln (10)} (\frac{1}{u_{2} \ln (10)} \frac{d}{d x} [100000^{200 x}])$

Step 2.3

Replace all occurrences of $u_{2}$ with $100000^{200 x}$ .

$\frac{1}{\log (100000^{200 x}) \ln (10)} (\frac{1}{100000^{200 x} \ln (10)} \frac{d}{d x} [100000^{200 x}])$

Step 3

Multiply $\frac{1}{100000^{200 x} \ln (10)}$ by $\frac{1}{\log (100000^{200 x}) \ln (10)}$ .

$\frac{1}{100000^{200 x} \ln (10) (\log (100000^{200 x}) \ln (10))} \frac{d}{d x} [100000^{200 x}]$

Step 4

Raise $\ln (10)$ to the power of $1$ .

$\frac{1}{100000^{200 x} (\ln^{1} (10) \ln (10)) \log (100000^{200 x})} \frac{d}{d x} [100000^{200 x}]$

Step 5

Raise $\ln (10)$ to the power of $1$ .

$\frac{1}{100000^{200 x} (\ln^{1} (10) \ln^{1} (10)) \log (100000^{200 x})} \frac{d}{d x} [100000^{200 x}]$

Step 6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{100000^{200 x} {\ln (10)}^{1 + 1} \log (100000^{200 x})} \frac{d}{d x} [100000^{200 x}]$

Step 7

Add $1$ and $1$ .

$\frac{1}{100000^{200 x} \ln^{2} (10) \log (100000^{200 x})} \frac{d}{d x} [100000^{200 x}]$

Step 8

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = 100000^{x}$ and $g (x) = 200 x$ .

Tap for more steps...

Step 8.1

To apply the Chain Rule, set $u_{3}$ as $200 x$ .

$\frac{1}{100000^{200 x} \ln^{2} (10) \log (100000^{200 x})} (\frac{d}{d u_{3}} [100000^{u_{3}}] \frac{d}{d x} [200 x])$

Step 8.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{3}} [a^{u_{3}}]$ is $a^{u_{3}} \ln (a)$ where $a$ = $100000$ .

$\frac{1}{100000^{200 x} \ln^{2} (10) \log (100000^{200 x})} (100000^{u_{3}} \ln (100000) \frac{d}{d x} [200 x])$

Step 8.3

Replace all occurrences of $u_{3}$ with $200 x$ .

$\frac{1}{100000^{200 x} \ln^{2} (10) \log (100000^{200 x})} (100000^{200 x} \ln (100000) \frac{d}{d x} [200 x])$

Step 9

Differentiate.

Tap for more steps...

Step 9.1

Combine $100000^{200 x}$ and $\frac{1}{100000^{200 x} \ln^{2} (10) \log (100000^{200 x})}$ .

$\frac{100000^{200 x}}{100000^{200 x} \ln^{2} (10) \log (100000^{200 x})} (\ln (100000) \frac{d}{d x} [200 x])$

Step 9.2

Simplify terms.

Tap for more steps...

Step 9.2.1

Combine $\ln (100000)$ and $\frac{100000^{200 x}}{100000^{200 x} \ln^{2} (10) \log (100000^{200 x})}$ .

$\frac{\ln (100000) \cdot 100000^{200 x}}{100000^{200 x} \ln^{2} (10) \log (100000^{200 x})} \frac{d}{d x} [200 x]$

Step 9.2.2

Cancel the common factor of $100000^{200 x}$ .

Tap for more steps...

Step 9.2.2.1

Cancel the common factor.

$\frac{\ln (100000) \cdot 100000^{200 x}}{100000^{200 x} \ln^{2} (10) \log (100000^{200 x})} \frac{d}{d x} [200 x]$

Step 9.2.2.2

Rewrite the expression.

$\frac{\ln (100000)}{\ln^{2} (10) \log (100000^{200 x})} \frac{d}{d x} [200 x]$

Step 9.3

Since $200$ is constant with respect to $x$ , the derivative of $200 x$ with respect to $x$ is $200 \frac{d}{d x} [x]$ .

$\frac{\ln (100000)}{\ln^{2} (10) \log (100000^{200 x})} (200 \frac{d}{d x} [x])$

Step 9.4

Combine $200$ and $\frac{\ln (100000)}{\ln^{2} (10) \log (100000^{200 x})}$ .

$\frac{200 \ln (100000)}{\ln^{2} (10) \log (100000^{200 x})} \frac{d}{d x} [x]$

Step 9.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{200 \ln (100000)}{\ln^{2} (10) \log (100000^{200 x})} \cdot 1$

Step 9.6

Multiply $\frac{200 \ln (100000)}{\ln^{2} (10) \log (100000^{200 x})}$ by $1$ .

$\frac{200 \ln (100000)}{\ln^{2} (10) \log (100000^{200 x})}$

Step 10

Simplify.

Tap for more steps...

Step 10.1

Expand $\log (100000^{200 x})$ by moving $200 x$ outside the logarithm.

$\frac{200 \ln (100000)}{\ln^{2} (10) (200 x \log (100000))}$

Step 10.2

Cancel the common factor of $200$ .

Tap for more steps...

Step 10.2.1

Cancel the common factor.

$\frac{200 \ln (100000)}{\ln^{2} (10) (200 x \log (100000))}$

Step 10.2.2

Rewrite the expression.

$\frac{\ln (100000)}{\ln^{2} (10) (x \log (100000))}$

Step 10.3

Logarithm base $10$ of $100000$ is $5$ .

$\frac{\ln (100000)}{\ln^{2} (10) x \cdot 5}$

Step 10.4

Move $5$ to the left of $\ln^{2} (10) x$ .

$\frac{\ln (100000)}{5 \cdot (\ln^{2} (10) x)}$

Step 10.5

Reorder factors in $\frac{\ln (100000)}{5 \ln^{2} (10) x}$ .

$\frac{\ln (100000)}{5 x \ln^{2} (10)}$